Animation with a slow start and quick end
Design an acceleration curve in the Graph editor.



Select the object that needs acceleration.
Open a Graph editor window. There will be lines plotting time against function (location, rotation, scale).
Select a keyframe point and press "v" on your keyboard. You get a dropdown menu that gives you vector behavior choices. (In my example, I selected my last keyframe, hit "v" and selected 'vector', which gives independent control over the incoming and outgoing bezier arms).
Adjust the function curve to give you the acceleration curve you're after.


Constant speed is a straight line.
Acceleration starts flattish on the left, and grows increasingly steeper.
Deceleration starts steep, then gradually flattens out.